By Richard Bellman (ed.)

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Fp ) is of finite type. If x is a point of V , we can find functions gi and h such that fi = gi /h, gi and h being regular in a neighborhood U of x and h being nonzero on U ; the sheaf R(f1 , . . , fp ) is then equal to the sheaf R(g1 , . . , gp ), which is of finite type, since OV is a coherent sheaf of rings. 41 Sheaf associated to the total space of a vector bundle Let E be an algebraic fiber space with a vector space of dimension r as a fiber and an algebraic variety V as a base; by definition, the typical fiber of E is a vector space K r and the structure group is the linear group GL(r, K) acting on K r in the usual way (for the definition of an algebraic fiber space, cf.

The space X being quasi-compact, we conclude that there exists a finite number of sections s1 , . . , sp of F generating Fx for all x ∈ X, which means that F is p isomorphic to a quotient sheaf of the sheaf OX . 54 §3. Coherent algebraic sheaves on affine varieties α II β Corollary 2. Let A − →B− → C be an exact sequence of coherent algebraic β α sheaf on an affine variety X. The sequence Γ(X, A ) − → Γ(X, B) − → Γ(X, C ) is also exact. We can suppose, as in the proof of Theorem 2, that X is an affine space K r , thus is irreducible.

D. (It is easily seen that this Proposition is false for prealgebraic varieties; the axiom (V AII ) plays an essential role). Let us now introduce a notation which will be used thorough the rest of this paragraph: if V is an algebraic variety and f is a regular function on V , we denote by Vf the open subset of V consisting of all points x ∈ V for which f (x) = 0. Proposition 2. If V is an affine algebraic variety and f is a regular function on V , the open subset Vf is affine. Let W be the subset of V ×K consisting of pairs (x, λ) such that λ·f (x) = 1; it is clear that W is closed in V ×K, thus it is an affine variety.

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